Heat Transfer Measurements on a Rotating Disk

Heat transfer to a rotating disk is measured for a wide range of Reynolds number values in 
the laminar, transitional and turbulent flow regimes. Measurements are performed by making 
use of the heated-thin-foil technique and by gauging temperature maps with an infrared 
scanning radiometer. The use of the IR radiometer is advantageous on account of its relatively 
good spatial resolution and thermal sensitivity and because it allows one to perform 
measurements down to very low local Reynolds numbers. Data is obtained on three disks, 
having an external diameter varying from 150mm to 450mm; the smallest disk is used only 
to measure the adiabatic wall temperature and can rotate up to 21,O00rpm. Heat transfer 
results are presented in terms of Nusselt and Reynolds numbers based on the local radius and 
show a substantial agreement with previous experimental and theoretical analyses. Transition 
to turbulent flow is found at about Re=250,000. A discussion about the role played by the adiabatic wall temperature is also 
included.


INTRODUCTION
The laminar flow due to an infinite fiat disk rotating in still air is one of the few exact solutions of the three-dimensional Navier-Stokes equations.This type of flow was first theoretically investigated with an approximate method by von KhrmS.n[1921] who found that it resembles a boundary layer flow but with a boundary layer thickness independent of the radial distance.The tangential component of the shear stress at the disk surface imparts a circumfer- ential velocity to the adjacent fluid layer which in turn, due to the centrifugal forces, also moves radi- ally outwards.Rogers and Lance [1960] calculated accurate solutions by means of a numerical integra- tion of the governing equations.Wagner [1948] first evaluated the convective heat transfer coefficient by finding an approximate solu- tion in the laminar regime based on the von Khrman velocity distribution.The  where a is a constant equal to 0.335 for Pr 0.74.Millsaps and Pohlhausen [1952] solved, still in the laminar regime, the exact equation of the thermal field by reducing the system of partial differential.*Corresponding author.
equations to an ordinary one by means of a similarity solution method.They imposed a boundary condition of constant temperature all over the disk and included the viscous dissipation effects.When the latter can be neglected and Pr 0.71, a is equal to 0.326.Cobb and Saunders [1956], by testing a disk rotat- ing from 30 to 2,500rpm, performed an experimental investigation on the mean heat transfer coefficient for a range of conditions from entirely laminar to mixed laminar-turbulent flow.In the laminar regime, they pointed out that eq. ( 1) is still valid when the average Nu and Re numbers are used.The lowest tested Reynolds number based on the disk radius is about 100,000.Although their data show a dependence of Nu from Re with an exponent which seems lower than 0.5, they affirmed that, in the laminar range, ex- perimental results fit eq. ( 1) with a coefficient a equal to 0.36.Moreover at the lowest tested Re, their find- ings are much larger than those predicted by eq. ( 1).
This may be due to the influence of the natural con- vection around the disk which becomes important at the lowest rotational speeds they tested.Cobb and   Saunders detected the onset of transition to turbulent flow at about Re 240,000.They also give the fol- lowing correlation between the local Nusselt and Reynolds numbers in the turbulent regime: Nu 0.0193 Re8 (2) which was obtained by means of the classic Reynolds analogy and the friction moment coefficient data of Theodorsen and Regier [1944].Kreith et al. [1959] experimentally evaluated mass transfer rates from a rotating disk of naphthalene un- der laminar and turbulent flow conditions and related their results to the heat transfer coefficients by means of an analogy method.They found a good agreement with theoretical data in the laminar range (a 0.34 for Pr 0.71) and located the onset of transition in the range 200,000 < Re < 250,000.
Popiel and Boguslawski [1975] measured the heat transfer coefficient at a certain location over a disk rotating at different angular speeds and found the co- efficient a of eq. ( 1) to be equal to 0.33 for laminar flow conditions.Also in their data the Nusselt number seems to depend on the Reynolds number with an exponent much lower than 0.5.Furthermore, at the lowest tested Re (7,000), Popiel and Boguslawski,   who also took into account the effects of natural con- vection, measured a Nusselt number which is about 35% larger than the theoretical prediction.They found the onset of transition at about Re 195,000.In the transitional and turbulent regimes their exper- imental data fit respectively the relationships: Nu 10. l0 -2.Re 4 for Re ranging from 195,000 to 250,000 and: (3) Nu 0.0188 Re '8  (4)   for Re greater than 250,000.It has to be pointed out that all the works men- tioned before deal with a boundary condition of iso- thermal disk which, in the laminar flow regime [see eq. ( 1)], coincides with the boundary condition of constant heat flux.For turbulent flow and for the boundary condition of a power law temperature dif- ference AT, between disk and surrounding fluid, Northrop and Owen [1988], according to the solution obtained by Dorfman 1963], presented the following correlation for the local Nusselt number: Nu 0.0197 (n + 2.6) +0.2 prO.6 ReO.8 (5) where n is the exponent of the power-law temperature difference profile" ATc r (6) and c is a constant.
The aim of this paper is to produce new data, in- cluding some at very low local Reynolds number and to discuss critically the validity limits of the previous correlations presented in the literature.Data about the adiabatic wall temperature behavior is also indicated for the laminar and turbulent flow regimes and in particular in the laminar regime data is compared with an analytical solution reported in the appendix.A major peculiarity of the present work lies in the fact that an infrared (IR) imaging system (IR scan- ning radiometer or thermograph) is employed to mea- HEAT TRANSFER MEASUREMENTS ON A ROTATING DISK sure the disk surface temperature, to evaluate the wall convective heat transfer coefficient.The advantage of having a non-contact temperature measuring device, such as the IR scanning radiometer, is well exploited to carry out heat transfer measurements in the vicin- ity of the disk axis of rotation, i.e. down to very low local Reynolds numbers.

EXPERIMENTAL APPARATUS AND PROCEDURE
A sketch of the experimental apparatus is shown in Fig 1 .The disk section consists of a 300ram (or 150mm or 450mm) in diameter aluminium (or light- alloy steel) cup filled with a 20mm thick polyurethane foam on which a printed circuit board is glued.The circuit is used to generate, by Joule effect, an uniform heat flux on the disk surface, while the polyurethane foam thermally insulates the face of the disk not ex- posed to air.Electric power is supplied to the printed circuit by means of a mercury rotating contact.The smallest disk, which is only used to measure adia- batic wall temperatures, is not heated.
A pulley, that is connected by a transmission belt to an electric motor, is fixed on the transmission shaft supporting the disk.The rotating speed of the disk, which may be positioned either perpendicular or parallel to the gravitational force, can be varied in a continuous way within the range O0-4,500rpm for the two larger disks and up to 21,000rpm for the smallest one.The disk angular speed is continuously monitored by a tachometer.It has to be pointed out that, in order to measure heat transfer coefficients at low local Reynolds numbers, one can either have the disk rotating at a low angular speed, or measure the temperature very close to the disk rotating axis.In the present tests the possibility of implementing the sec- ond procedure makes it possible to keep relatively high rotating speed so as to diminish natural convec- tion effects.
The printed circuit board is designed so as to achieve a constant heat flux over the disk surface and therefore the thickness and the width of its conduct- ing tracks, having a spiral shape, are manufactured with very close tolerances.Tracks are 35am thick, 2mm wide and placed at 2.5mm pitch; the overall thickness of the board is 0.2mm.Details about the printed circuit are reported in Cardone et al. [1993].
To enhance the emitted IR radiation detection, the measured board surface is coated with a thin layer of black paint which has an emissivity coefficient equal to 0.95 in the working IR window of the employed scanner.
The infrared thermographic system is based on AGEMA Thermovision 880.The field of view (which depends on the optic focal length and on the viewing distance) is scanned by the Hg-Cd-Te detector in the 8-12am IR window.Nominal sensitivity, expressed in terms of noise equivalent temperature difference is 0.1 C when the scanned object is at ambient tempera- ture.The scanning spatial resolution is 175 instanta- neous fields of view per line at 50% slit response func- tion.A 20 20 lens is used during the tests.The thermal image is digitized in a frame of 140 140 pixels 8 bits.In order to achieve an azimuthal spa- tial resolution the line scan facility of AGEMA 880 IR camera is also used to take temperature radial profiles (see Cardone et al. [1993]).
where qi is the Joule heating, qr, is the radiative heat flux to ambient, T is the measured wall temperature and Tw the adiabatic wall temperature of the flow.
The radiative thermal losses qr, are computed by us- ing the measured T while the conductive ones to- ward the inner polyurethane foam, are neglected.Taw is measured by means of the same thermographic technique under the assumption that it coincides with the disk surface temperature when the Joule heating is suppressed; in effects a small correction of the or- der of few percent has to be made to take into ac- count thermal radiation effects.The role played by the use of T,w in eq. ( 7) in place of the ambient temperature T,, will be discussed later.
An error analysis, based on calibration accuracy of infrared system and repeatability of measurements in- dicates that heat transfer coefficients measurements are accurate to within about ___3%.

RESULTS
FIGURE 2 Thermogram of the 300ram disk at 576rpm and qi 407 W/m Fig. 3 is a thermal picture of the largest disk (450mm) recorded while it is rotating at 4390rpm and is subject to a heat flux of 871 W/me.A relatively small (about 16% of the total surface) region around the disk center, where the wall temperature is con- stant, is clearly evident.On the basis of the previous discussion the flow is laminar there.In the outer zone the temperature decreases, first quickly in the transi- tional regime and then slowly in the turbulent one; In Fig 2 a thermographic image of the disk (D 300mm), .rotatingat 576rpm and subject to a heat flux qi 407W/m , is shown.Due to the relatively low angular speed, the flow is expected to be laminar ev- erywhere over the disk.According to this, as shown by the thermogram, the disk surface exhibits a uni- form wall temperature which in the present case is about 45C.In fact, the laminar correlation of eq. ( 1) predicts the h coefficient to be constant over the disk; as a consequence, from eq. ( 7) it arises that, for a uniform heat flux boundary condition and uniform distribution of Tw (in the low subsonic regime Tw coincides with the ambient temperature), the wall temperature must be constant.Near the periphery of the disk a temperature decrease, due to edge effects, is found.Thermogram of the 450ram disk at 4390rpm and qi HEAT TRANSFER MEASUREMENTS ON A ROTATING DISK immediately after the temperature trend is reversed as T slowly begins to rise.Also in this case edge effects appear near the disk periphery.
In order to explain the temperature behavior in the turbulent regime, first consider that due to the turbu- lent correlation law [e.g.eq. ( 5)] the heat transfer coefficient is expected to increase as the local radius increases.As long as the adiabatic wall temperature distribution is uniform eq. ( 7) shows that-the wall temperature must decrease along the radial direction.However, by examining the cold thermogram of Fig. 4, which is relative to the adiabatic wall temperature recorded at the same disk angular speed as that of Fig. 3, it should be noted that T,w is practically con- stant (and equal to T,) only within the circumference whose radius is about 60% of that of the disk.After- words T,, experiences a significant increase (about 3C over T,, near the disk edge).Since for the present experimental conditions T T,w is of the same order of magnitude as T,, T,, the increasing trend of the wall temperature shown in Fig. 3 is accordingly ex- plained.It has to be explicitly pointed out that in the case of relatively high boundary heat fluxes qi (i.e. high AT), the effect of the adiabatic wall temperature becomes negligible and T is monotonically decreas- ing towards the disk periphery.The thermogram of Fig. 5, showing the AT map, may be interpreted as an overall picture of the heat transfer coefficient surface distribution.The sequence of annular rings, which in- creasingly darkens towards the disk limb, proves the qualitative trend of the h coefficient.
The temperature difference T,w T, for the un- heated 150mm disk is plotted in Fig. 6 for angular speeds varying from 11,900 to 20,600rpm..Each tem- perature profile is the average of data relative to three different tests which shows a very small spread (<1%).In the same figure the line corresponding to the onset of transitional and turbulent flows are also shown.
Since the recovery factor is nearly constant throughout the disk, the temperature profile is practi- cally parabolic.In fact the recovery factor in the lam- inar regime, as computed from data of Fig. 6, is in accordance with the theoretical value indicated in the appendix which, for Pr 0.71, gives R 0.891.The recovery factor in the transitional and turbulent flow regimes is not very different since it varies from 0.886 to 0.894 which is very close to /r 0.892.It has to be pointed out however, that since the temperature differences are small and the resolu- tion of the scanner is equal to 0.1C, no definitive statement can be made for the laminar flow regime.
Furthermore it must be recalled that the R values are computed by considering also the losses for thermal radiation which account for a correction of a few percent.
Data obtained in laminar flow regime, with both the 300ram and 450mm in diameter disks rotating at 34 different angular speeds, are shown in Fig. 7 in terms of local Nusselt number for 4 < Re < 200, 000.
The theoretical prediction of Millsaps and Pohl-  hausen [1952] is also reported.All points fall around a straight line in the log-log plane down to very low Reynolds numbers.Following the Wagner [1948] the- ory, a correlation of all the data in terms of equation ( 1) is made and the value of the constant results a 0.333.This value looks to be very much in accor- dance with previous theoretical and experimental findings.It has to be stressed that the validity of (1) down to very low local Reynolds numbers has not been proved before.
To check the actual slope of data in the log-log plane, a generalization of Wagner [1948]  A linear regression based on eq. ( 8) is applied to a sample of data, first by neglecting Taw effects and then by taking them into account.In the former case the evaluated slope is b 0.488, in the latter one b 0.499.This finding demonstrates that the not very accurate slope shown-by data of previous investiga- tors may be attributed not only to the natural convec- tion effects but also to the Taw effect which was sys- tematically neglected.
The results relative to the 450mm-in diameter disk for Re ranging from 1,000 to 1,400,000, are shown in Fig. 8.While the almost sudden rise of Nu around Re 250,000 is to be ascribed to the onset of transition, the second slope change, which appears at the right end of the figure, is be to attributed to the presence of fully turbulent flow.A linear regression of the data in the transitional range of Re from 260,000 to 320,000 yields: HEAT TRANSFER MEASUREMENTS ON A ROTATING DISK Present transitional results do not agree with those of Popiel and Boguslawski [1975] as far as both the transitional Re range and the regression slope are concerned.Apart from the remark that measurements in transitional flow regimes may be in general strongly affected by the environmental conditions of the actual experimental apparatus, it should be stressed that the results of Popiel and Boguslawski [1975] are relative to an isothermal boundary condi- tion and are obtained by using a calorimetric device too large to achieve the fine spatial resolution exhib- ited by the data reported in their paper.
In the fully turbulent regime present data fits the relation: Nu 0.0163.Re 's (10) According to (5) the convective heat transfer coeffi- cient is proportional to 2 0.6.In the case of a constant heat flux boundary condition, AT is inversely propor- tional to h so that the coefficient n is equal to -0.6; therefore, for Pr .71eq. ( 5) reduces to the relation- ship: Nu 0.0184 Re (11) which is quite in accordance with the present experimental data.
Finally it has to be said that the obtained experimental results are practically identical for both disks perpendicular to the direction of the gravitational force and disks parallel to this latter and facing down- ward; this behavior confirms that the neglecting of natural convection effects, for the tested rotational speeds, is correct.its relatively good spatial resolution and thermal sen- sitivity and because it facilitates the making of mea- surements down to very low local Reynolds num- bers.
Experimental data is correlated in terms of Nu and Re numbers, both based on the local radius and shows to be in accordance with theoretical predictions for a range of Reynolds numbers much wider than that obtained in previous studies.Moreover, in the authors' opinion, transitional results seem to be more reliable than the ones available in the literature.
In particular, the onset of the transition to turbulence is found around Re 250,000.
A theoretical discussion about the role played by the adiabatic wall temperature in the laminar regime is made.The analysis seems to be confirmed by ex- perimental results.In the turbulent regime the fact that the adiabatic wall temperature rises towards the disk limb gives an explanation of the increase of the heated disk surface temperature in that region.
APPENDIX A cylindirical coordinate system is assumed, as shown in Fig. 9.To reduce the Navier Stokes equa- tions to a dimensionless ordinary differential system, von Khrmhn [1921] imposed the pertinent variables to satisfy the relations: The boundary conditions reduce to: F*(0) 0; HF*(w) 0; HG*(O) 1; HG*(w) 0; H*(0) 0; (14) Millsaps and Pohlhausen [1952] proposed, for the thermal field, the following relations: T (veolcp) T* T* Re S*(z*) + Q*(z*) + T* (15) so that the governing equations of the thermal field are: S*" Pr H'S*' + Pr H*'S* Pr (/7*'2 q-G*'2) Q*" Pr H'Q*' -(4 S* + 12 Pr F.2) (16) They imposed a constant temperature boundary con- dition over the disk.Herein a constant heat flux and particularly the adiabatic wall condition, OT/Ozlw 0 is imposed, which implies that on the disk surface: S*'(0) 0; Q*'(0) 0 (17) while at a large distance from the disk the boundary condition of the constant ambient temperature im- posed by Millsaps and Pohlhausen still holds: S*(c) 0; Q,(c) 0 (18) The variation with.r of the wall temperature is due only to S: T:aw Re2 S*(0) + T*a (19) and, by returning to dimensional quantity, the follow- ing is obtained: The recovery factor which is defined as: R (21) (r00) 2 2Cp in the flow laminar regime turn out to be equal to 2. s(0).Table reports the recovery factor, obtained by solving the system of differential equations ( 13) and ( 16) with the boundary conditions ( 14), ( 17) and ( 18), as a function of the Prandtl number.

FIGURE 4
FIGURE 4 Thermogram of the 450mm disk at 4390rpm and qi OW/me.

FIGURE 7
FIGURE 7 Nusselt number as a function of Reynolds number.FIGURE 8 Nusselt number as a function of Reynolds number.

FIGURE 9
FIGURE 9 Sketch of the coordinate system.G*"-H* G*' + 2F* G 0 P*' H*" H'H*' 0 (13) ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy SystemsEconomic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y

TABLE
Recovery Factor as a Function of Pr.